<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon">M</font> =  <a href="rlvect_1.html#G1">LoopStr</a>(# <font color="Maroon">I</font>,<span class="p1">(<span class="default"><a href="ideal_1.html#K1" target="_self">add|</a> <font color="Maroon">I</font>,<font color="Maroon">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="funct_7.html#K1">In</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">I</font></span>)</span> #);<br/>
<b>reconsider </b><font color="Maroon">M</font> =  <a href="rlvect_1.html#G1">LoopStr</a>(# <font color="Maroon">I</font>,<span class="p1">(<span class="default"><a href="ideal_1.html#K1" target="_self">add|</a> <font color="Maroon">I</font>,<font color="Maroon">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="funct_7.html#K1">In</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">I</font></span>)</span> #) as  non <a href="struct_0.html#V3">empty</a>  <a href="rlvect_1.html#L1">LoopStr</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:39_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">I</font>,<font color="Maroon">I</font></span><a href="mcart_1.html#K12">:]</a></span>
 ;<br/><a NAME="E2:39_1_1_1"/>
 <a href="relset_1.html#K4">dom</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E3:39_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E3:39_1_1"/><b>then </b>
<span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">I</font>,<font color="Maroon">I</font></span><a href="mcart_1.html#K12">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:39_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font> <a href="realset1.html#K1">||</a> <font color="Maroon">I</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">I</font>,<font color="Maroon">I</font></span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E5:39_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font><br/> for <font color="Olive">a'</font>, <font color="Olive">b'</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">I</font>  st <font color="Olive">a'</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> holds <br/><font color="Olive">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">a'</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">a'</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">I</font>, <font color="Maroon">b'</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">I</font>;<br/>



<b>assume </b><a NAME="E1:39_1_1_2"/>
( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 ;<br/>

<b>hence </b><font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b</font> = 
<span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font> <a href="realset1.html#K1">||</a> <font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a'</font>,<font color="Maroon">b'</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a'</font>,<font color="Maroon">b'</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">a'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b'</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
;<br/><br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">I</font> ;<br/><b>thus </b><font color="Maroon">a</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b</font> = 
<font color="Maroon">a'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">b</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">a</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:39_1_1"/>
 <a href="ideal_1.html#K3" target="_self">Gr</a> <font color="Maroon">I</font>,<font color="Maroon">R</font> is <a href="rlvect_1.html#V3">Abelian</a>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D5">RLVECT_1:def 5</a></i>;<br/>


</div>
